Sorry I meant when I was trying rotate no.2 nut, whole pipe is rotating.

How many wrenches are you using? One high, one low,counter rotating.

Thank you @A Mat. So, is nut#2, the correct nut to disconnect when installing a new faucet? I was only using one wrench - I will try with two wrenches. Thanks for the suggestion.

Also,

1. I turned off the main water valve, checked that none of the kitchen faucets had running water, then came to bathroom and turned on the shower faucet, some water dripped for at least 3 minutes and then finally there was no water in bathroom. However, when I loosened the nut #1, water came dripping for quite some time, is it residual water but then I tightened the nut again. Why would the shower and faucet have no water but the pipe near nut #1 had water?

2. Is it possible that there could be another water main connection that I need to turn off? Is it common for a condo to have two water shutoffs?

You can remove either the top flexible hose or the short extension below that is attached to the valve. It depends how long your new faucet supply flex tubing is.
There is some residual water in the faucet. Just put a towel around the bottom of the pipe when you loose it to absorb the water. Happens all the time.
Be sure you have closed the valves right there at the wall inside the vanity. With them closed you shouldn’t need to turn off the main water valve.

Thank you @BobH When you say there will be some residual water - would there be water even after I verified that the bath tub and sink faucet both did not have water prior to loosening the nut?

Update: I turned off the main shutoff to the unit and tried loosening #2 supply line. There was a steady steam of water dripping from the faucet supply line (note: not the copper pipe but faucet's flexible line). I thought it must be residual water and by that time water everywhere else stopped. I let the water drip for 20 more minutes and by that time I was able to fill 15 gallon water bucket easily. I didn't find it encouraging so I aborted my plan. Stream was steady even after 20 minutes. How can there be so much residual water? I double checked and water was turned off at main shutoff. Can someone give me pointers on where the water is coming from?

Did you turn off the valve right there below #1 nut in your picture? If not then water was draining from any fixtures that are higher than the nut you loosened. (2nd floor fixtures/shower valve, etc.)

Thank you @BobH. Water is coming out of flexible supply line attached to faucet (supply line above #2) and not the copper pipe above #1 and below #2) I started by loosening the #2 which is hot water line but steady water coming was cold. I didn't loosen or touch the cold water line. I hope I am making sense. Also, this is a single level condo.

Did you close both the hot and cold valves (below nut #1) before you removed the flex line?

Thanks @BobH. I didn't turn off the valves underneath the sink but I turned off the main shutoff.

15 gallon water bucket? That would weigh ~ 120 lbs. full. If it was indeed 15 gallons, I doubt draining your lines would produce that much water. Certainly not the case in my house. Could be your main shutoff is leaking through. How old is the plumbing and what sort of valve is the shutoff?

Sorry @sofaspud I meant 15 gallons in a bucket. How can I tell that main shutoff is leaking? Wouldn't I see water? I am not sure how old the plumbing is - gosh, that is a very good question. I purchased this condo recently. Is there a way to confirm that?

@BobH I am unable to rotate the valves underneath the sink - they are stuck and that is the reason I have been using main shutoff.

Ok. If you can’t shut off the valves under the vanity then close the main water valve and open this faucet where you are working. Go to all your other faucets throughout the house and open them. Any water that is higher than the faucet you are working on will drain out your faucet. If you have a faucet that is lower than the one you are working on that is good news and the water will drain to that point. Once the water has stopped dripping out of your faucet have your bucket under the vanity and loosen the nut. Have a big towel there too to absorb water. You should not get nearly as much water as you got last time. If you have a faucet open that is lower than the one you are working on and your main valve is leaking then that faucet will continue to drip. If the one you are working on is the lowest then either try to close the valve under the vanity with a wrench of some sort or just let it drip into your towel and replace your faucet quickly.

Valve leaking through means it's shut but the internal seals are leaking, so water gets into the lines regardless. That's why I asked how old and what type. Can't walk you through replacing it.

Hello @BobH and @sofaspud. I have an update. I turned off the main valve and this time I also turned off hot water tank's valve. Went to the bathroom faucet sink and slowly loosened the compression nut for supply lines so that I could replace the faucet. After I loosened the hot line, there was no water coming from the line which was a good sign. I proceeded to loosen the cold water line but there was steady stream of water coming out of it, it was low pressure stream.

I took the chance and connected new faucet. During this process I was able to shut off the valves underneath the sink and that stopped the steady stream of water. I completed my faucet installation successfully. I turned on the water, tested the faucet and there were no leaks. Only thing I noticed is that the new faucet is dripping at a slow rate like 2 drops per minute.

1. What could be causing slow drips from new faucet, for that matter even old faucet used to drip.
2. And, why was the cold water line not shut off completely even though I turned off the main line. Are they connected? Could I have a leaking plumbing pipe somewhere in the system?

I wanted to thank all of you 🙏 for helping me out during this journey. Mission accomplished but I am curious about those two questions.